ILL 1990 Problem 54

Let $M = \{1, 2, \ldots, n\}$ and $\phi : M \to M$ be a bijection.

(i) Prove that there exist bijections $\phi_1, \phi_2 : M \to M$ such that $\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E$ , where $E$ is the identity mapping.

(ii) Prove that the conclusion in (i) is also true if $M$ is the set of all positive integers.